2frac (problem 3.3.1)

Percentage Accurate: 77.7% → 98.9%

Time: 8.6s

Alternatives: 11

Speedup: 0.7×

Specification

Math

\[\begin{array}{l} \\ \frac{1}{x + 1} - \frac{1}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))

C

double code(double x) {
	return (1.0 / (x + 1.0)) - (1.0 / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (x + 1.0d0)) - (1.0d0 / x)
end function

Java

public static double code(double x) {
	return (1.0 / (x + 1.0)) - (1.0 / x);
}

Python

def code(x):
	return (1.0 / (x + 1.0)) - (1.0 / x)

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(1.0 / x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (x + 1.0)) - (1.0 / x);
end

Wolfram

code[x_] := N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Initial Program: 77.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + 1} - \frac{1}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))

C

double code(double x) {
	return (1.0 / (x + 1.0)) - (1.0 / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (x + 1.0d0)) - (1.0d0 / x)
end function

Java

public static double code(double x) {
	return (1.0 / (x + 1.0)) - (1.0 / x);
}

Python

def code(x):
	return (1.0 / (x + 1.0)) - (1.0 / x)

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(1.0 / x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (x + 1.0)) - (1.0 / x);
end

Wolfram

code[x_] := N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x} + \frac{-1}{x}\\ \mathbf{if}\;t\_0 \leq -500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(1 + \frac{1}{x \cdot x}\right) \cdot \frac{1}{\left(x \cdot x\right) \cdot \frac{x}{1 - x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x))))
   (if (<= t_0 -500.0)
     t_0
     (if (<= t_0 0.0)
       (* (+ 1.0 (/ 1.0 (* x x))) (/ 1.0 (* (* x x) (/ x (- 1.0 x)))))
       t_0))))

C

double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (1.0 + (1.0 / (x * x))) * (1.0 / ((x * x) * (x / (1.0 - x))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / (1.0d0 + x)) + ((-1.0d0) / x)
    if (t_0 <= (-500.0d0)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = (1.0d0 + (1.0d0 / (x * x))) * (1.0d0 / ((x * x) * (x / (1.0d0 - x))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (1.0 + (1.0 / (x * x))) * (1.0 / ((x * x) * (x / (1.0 - x))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x)
	tmp = 0
	if t_0 <= -500.0:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (1.0 + (1.0 / (x * x))) * (1.0 / ((x * x) * (x / (1.0 - x))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-1.0 / x))
	tmp = 0.0
	if (t_0 <= -500.0)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(1.0 + Float64(1.0 / Float64(x * x))) * Float64(1.0 / Float64(Float64(x * x) * Float64(x / Float64(1.0 - x)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	tmp = 0.0;
	if (t_0 <= -500.0)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (1.0 + (1.0 / (x * x))) * (1.0 / ((x * x) * (x / (1.0 - x))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -500.0], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(1.0 + N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(x / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < -500 or 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   if -500 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 59.2%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{x} + \frac{1}{{x}^{3}}\right) - \left(1 + \frac{1}{{x}^{2}}\right)}{{x}^{2}}} \]

   5. Applied rewrites 84.3%

      \[\leadsto \color{blue}{\left(1 + \frac{1}{x \cdot x}\right) \cdot \frac{1 - x}{x \cdot \left(x \cdot x\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.6%

      \[\leadsto \left(1 + \frac{1}{x \cdot x}\right) \cdot \frac{1}{\color{blue}{\frac{x}{1 - x} \cdot \left(x \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} + \frac{-1}{x} \leq -500:\\ \;\;\;\;\frac{1}{1 + x} + \frac{-1}{x}\\ \mathbf{elif}\;\frac{1}{1 + x} + \frac{-1}{x} \leq 0:\\ \;\;\;\;\left(1 + \frac{1}{x \cdot x}\right) \cdot \frac{1}{\left(x \cdot x\right) \cdot \frac{x}{1 - x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x} + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x} + \frac{-1}{x}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1 + \frac{x + -1}{x \cdot x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x))))
   (if (<= t_0 -1e-8)
     t_0
     (if (<= t_0 0.0) (/ (+ -1.0 (/ (+ x -1.0) (* x x))) (* x x)) t_0))))

C

double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -1e-8) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 + ((x + -1.0) / (x * x))) / (x * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / (1.0d0 + x)) + ((-1.0d0) / x)
    if (t_0 <= (-1d-8)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = ((-1.0d0) + ((x + (-1.0d0)) / (x * x))) / (x * x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -1e-8) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 + ((x + -1.0) / (x * x))) / (x * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x)
	tmp = 0
	if t_0 <= -1e-8:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (-1.0 + ((x + -1.0) / (x * x))) / (x * x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-1.0 / x))
	tmp = 0.0
	if (t_0 <= -1e-8)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-1.0 + Float64(Float64(x + -1.0) / Float64(x * x))) / Float64(x * x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	tmp = 0.0;
	if (t_0 <= -1e-8)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (-1.0 + ((x + -1.0) / (x * x))) / (x * x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-8], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(-1.0 + N[(N[(x + -1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < -1e-8 or 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x))

   1. Initial program 99.9%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   if -1e-8 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 59.0%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{x} - \left(1 + \frac{1}{{x}^{2}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} - 1\right) - \frac{1}{{x}^{2}}}}{{x}^{2}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{x} - 1\right) + \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)}}{{x}^{2}} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right) + \left(\frac{1}{x} - 1\right)}}{{x}^{2}} \]

      4. associate-+r- N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right) + \frac{1}{x}\right) - 1}}{{x}^{2}} \]

      5. neg-sub0 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(0 - \frac{1}{{x}^{2}}\right)} + \frac{1}{x}\right) - 1}{{x}^{2}} \]

      6. associate--r- N/A

         \[\leadsto \frac{\color{blue}{\left(0 - \left(\frac{1}{{x}^{2}} - \frac{1}{x}\right)\right)} - 1}{{x}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(0 - \left(\frac{1}{\color{blue}{x \cdot x}} - \frac{1}{x}\right)\right) - 1}{{x}^{2}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{\left(0 - \left(\color{blue}{\frac{\frac{1}{x}}{x}} - \frac{1}{x}\right)\right) - 1}{{x}^{2}} \]

      9. div-sub N/A

         \[\leadsto \frac{\left(0 - \color{blue}{\frac{\frac{1}{x} - 1}{x}}\right) - 1}{{x}^{2}} \]

      10. neg-sub0 N/A

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{x} - 1}{x}\right)\right)} - 1}{{x}^{2}} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\frac{1}{x} - 1}{x}} - 1}{{x}^{2}} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-1 \cdot \frac{\frac{1}{x} - 1}{x} - 1}{{x}^{2}}} \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{-1 + \frac{-1 + x}{x \cdot x}}{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} + \frac{-1}{x} \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{1 + x} + \frac{-1}{x}\\ \mathbf{elif}\;\frac{1}{1 + x} + \frac{-1}{x} \leq 0:\\ \;\;\;\;\frac{-1 + \frac{x + -1}{x \cdot x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x} + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x} + \frac{-1}{x}\\ \mathbf{if}\;t\_0 \leq -500:\\ \;\;\;\;\left(x + -1\right) \cdot \frac{\mathsf{fma}\left(x, x, 1\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, x, 1\right) - x\right) + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x))))
   (if (<= t_0 -500.0)
     (* (+ x -1.0) (/ (fma x x 1.0) x))
     (if (<= t_0 0.0) (/ (/ -1.0 x) x) (+ (- (fma x x 1.0) x) (/ -1.0 x))))))

C

double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = (x + -1.0) * (fma(x, x, 1.0) / x);
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 / x) / x;
	} else {
		tmp = (fma(x, x, 1.0) - x) + (-1.0 / x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-1.0 / x))
	tmp = 0.0
	if (t_0 <= -500.0)
		tmp = Float64(Float64(x + -1.0) * Float64(fma(x, x, 1.0) / x));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-1.0 / x) / x);
	else
		tmp = Float64(Float64(fma(x, x, 1.0) - x) + Float64(-1.0 / x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -500.0], N[(N[(x + -1.0), $MachinePrecision] * N[(N[(x * x + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(x * x + 1.0), $MachinePrecision] - x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < -500

   1. Initial program 100.0%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 + x \cdot \left(x - 1\right)\right) - 1}{x}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot \left(1 + x \cdot \left(x - 1\right)\right)}{x} - \frac{1}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(x - 1\right)\right) \cdot x}}{x} - \frac{1}{x} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(1 + x \cdot \left(x - 1\right)\right) \cdot \frac{x}{x}} - \frac{1}{x} \]

      4. *-inverses N/A

         \[\leadsto \left(1 + x \cdot \left(x - 1\right)\right) \cdot \color{blue}{1} - \frac{1}{x} \]

      5. *-rgt-identity N/A

         \[\leadsto \color{blue}{\left(1 + x \cdot \left(x - 1\right)\right)} - \frac{1}{x} \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x - 1\right) + 1\right)} - \frac{1}{x} \]

      7. associate--l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(x - 1\right) + \left(1 - \frac{1}{x}\right)} \]

      8. sub-neg N/A

         \[\leadsto x \cdot \left(x - 1\right) + \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto x \cdot \left(x - 1\right) + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) + 1\right)} \]

      10. neg-sub0 N/A

          \[\leadsto x \cdot \left(x - 1\right) + \left(\color{blue}{\left(0 - \frac{1}{x}\right)} + 1\right) \]

      11. associate-+l- N/A

          \[\leadsto x \cdot \left(x - 1\right) + \color{blue}{\left(0 - \left(\frac{1}{x} - 1\right)\right)} \]

      12. neg-sub0 N/A

          \[\leadsto x \cdot \left(x - 1\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{x} - 1\right)\right)\right)} \]

      13. unsub-neg N/A

          \[\leadsto \color{blue}{x \cdot \left(x - 1\right) - \left(\frac{1}{x} - 1\right)} \]

      14. *-inverses N/A

          \[\leadsto x \cdot \left(x - 1\right) - \left(\frac{1}{x} - \color{blue}{\frac{x}{x}}\right) \]

      15. div-sub N/A

          \[\leadsto x \cdot \left(x - 1\right) - \color{blue}{\frac{1 - x}{x}} \]

      16. unsub-neg N/A

          \[\leadsto x \cdot \left(x - 1\right) - \frac{\color{blue}{1 + \left(\mathsf{neg}\left(x\right)\right)}}{x} \]

      17. mul-1-neg N/A

          \[\leadsto x \cdot \left(x - 1\right) - \frac{1 + \color{blue}{-1 \cdot x}}{x} \]

      18. *-rgt-identity N/A

          \[\leadsto x \cdot \left(x - 1\right) - \frac{\color{blue}{\left(1 + -1 \cdot x\right) \cdot 1}}{x} \]

      19. associate-/l* N/A

          \[\leadsto x \cdot \left(x - 1\right) - \color{blue}{\left(1 + -1 \cdot x\right) \cdot \frac{1}{x}} \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\left(-1 + x\right) \cdot \left(x + \frac{1}{x}\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(-1 + x\right) \cdot \frac{1 + {x}^{2}}{\color{blue}{x}} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.6%

      \[\leadsto \left(-1 + x\right) \cdot \frac{\mathsf{fma}\left(x, x, 1\right)}{\color{blue}{x}} \]

   if -500 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 59.2%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-1}{\color{blue}{x \cdot x}} \]

      3. lower-*.f64 98.2

         \[\leadsto \frac{-1}{\color{blue}{x \cdot x}} \]

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\frac{-1}{x \cdot x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.5%

      \[\leadsto \frac{\frac{-1}{x}}{\color{blue}{x}} \]

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x - 1\right)\right)} - \frac{1}{x} \]
   4. Step-by-step derivation

      1. distribute-rgt-out-- N/A

         \[\leadsto \left(1 + \color{blue}{\left(x \cdot x - 1 \cdot x\right)}\right) - \frac{1}{x} \]

      2. unpow2 N/A

         \[\leadsto \left(1 + \left(\color{blue}{{x}^{2}} - 1 \cdot x\right)\right) - \frac{1}{x} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \left(1 + \color{blue}{\left({x}^{2} + \left(\mathsf{neg}\left(1\right)\right) \cdot x\right)}\right) - \frac{1}{x} \]

      4. metadata-eval N/A

         \[\leadsto \left(1 + \left({x}^{2} + \color{blue}{-1} \cdot x\right)\right) - \frac{1}{x} \]

      5. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + {x}^{2}\right) + -1 \cdot x\right)} - \frac{1}{x} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(1 + {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) - \frac{1}{x} \]

      7. unsub-neg N/A

         \[\leadsto \color{blue}{\left(\left(1 + {x}^{2}\right) - x\right)} - \frac{1}{x} \]

      8. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + {x}^{2}\right) - x\right)} - \frac{1}{x} \]

      9. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left({x}^{2} + 1\right)} - x\right) - \frac{1}{x} \]

      10. unpow2 N/A

          \[\leadsto \left(\left(\color{blue}{x \cdot x} + 1\right) - x\right) - \frac{1}{x} \]

      11. lower-fma.f64 99.1

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(x, x, 1\right)} - x\right) - \frac{1}{x} \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) - x\right)} - \frac{1}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} + \frac{-1}{x} \leq -500:\\ \;\;\;\;\left(x + -1\right) \cdot \frac{\mathsf{fma}\left(x, x, 1\right)}{x}\\ \mathbf{elif}\;\frac{1}{1 + x} + \frac{-1}{x} \leq 0:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, x, 1\right) - x\right) + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x} + \frac{-1}{x}\\ \mathbf{if}\;t\_0 \leq -500:\\ \;\;\;\;\left(x + -1\right) \cdot \left(x + \frac{1}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, x, 1\right) - x\right) + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x))))
   (if (<= t_0 -500.0)
     (* (+ x -1.0) (+ x (/ 1.0 x)))
     (if (<= t_0 0.0) (/ (/ -1.0 x) x) (+ (- (fma x x 1.0) x) (/ -1.0 x))))))

C

double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = (x + -1.0) * (x + (1.0 / x));
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 / x) / x;
	} else {
		tmp = (fma(x, x, 1.0) - x) + (-1.0 / x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-1.0 / x))
	tmp = 0.0
	if (t_0 <= -500.0)
		tmp = Float64(Float64(x + -1.0) * Float64(x + Float64(1.0 / x)));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-1.0 / x) / x);
	else
		tmp = Float64(Float64(fma(x, x, 1.0) - x) + Float64(-1.0 / x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -500.0], N[(N[(x + -1.0), $MachinePrecision] * N[(x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(x * x + 1.0), $MachinePrecision] - x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < -500

   1. Initial program 100.0%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 + x \cdot \left(x - 1\right)\right) - 1}{x}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot \left(1 + x \cdot \left(x - 1\right)\right)}{x} - \frac{1}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(x - 1\right)\right) \cdot x}}{x} - \frac{1}{x} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(1 + x \cdot \left(x - 1\right)\right) \cdot \frac{x}{x}} - \frac{1}{x} \]

      4. *-inverses N/A

         \[\leadsto \left(1 + x \cdot \left(x - 1\right)\right) \cdot \color{blue}{1} - \frac{1}{x} \]

      5. *-rgt-identity N/A

         \[\leadsto \color{blue}{\left(1 + x \cdot \left(x - 1\right)\right)} - \frac{1}{x} \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x - 1\right) + 1\right)} - \frac{1}{x} \]

      7. associate--l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(x - 1\right) + \left(1 - \frac{1}{x}\right)} \]

      8. sub-neg N/A

         \[\leadsto x \cdot \left(x - 1\right) + \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto x \cdot \left(x - 1\right) + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) + 1\right)} \]

      10. neg-sub0 N/A

          \[\leadsto x \cdot \left(x - 1\right) + \left(\color{blue}{\left(0 - \frac{1}{x}\right)} + 1\right) \]

      11. associate-+l- N/A

          \[\leadsto x \cdot \left(x - 1\right) + \color{blue}{\left(0 - \left(\frac{1}{x} - 1\right)\right)} \]

      12. neg-sub0 N/A

          \[\leadsto x \cdot \left(x - 1\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{x} - 1\right)\right)\right)} \]

      13. unsub-neg N/A

          \[\leadsto \color{blue}{x \cdot \left(x - 1\right) - \left(\frac{1}{x} - 1\right)} \]

      14. *-inverses N/A

          \[\leadsto x \cdot \left(x - 1\right) - \left(\frac{1}{x} - \color{blue}{\frac{x}{x}}\right) \]

      15. div-sub N/A

          \[\leadsto x \cdot \left(x - 1\right) - \color{blue}{\frac{1 - x}{x}} \]

      16. unsub-neg N/A

          \[\leadsto x \cdot \left(x - 1\right) - \frac{\color{blue}{1 + \left(\mathsf{neg}\left(x\right)\right)}}{x} \]

      17. mul-1-neg N/A

          \[\leadsto x \cdot \left(x - 1\right) - \frac{1 + \color{blue}{-1 \cdot x}}{x} \]

      18. *-rgt-identity N/A

          \[\leadsto x \cdot \left(x - 1\right) - \frac{\color{blue}{\left(1 + -1 \cdot x\right) \cdot 1}}{x} \]

      19. associate-/l* N/A

          \[\leadsto x \cdot \left(x - 1\right) - \color{blue}{\left(1 + -1 \cdot x\right) \cdot \frac{1}{x}} \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\left(-1 + x\right) \cdot \left(x + \frac{1}{x}\right)} \]

   if -500 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 59.2%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-1}{\color{blue}{x \cdot x}} \]

      3. lower-*.f64 98.2

         \[\leadsto \frac{-1}{\color{blue}{x \cdot x}} \]

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\frac{-1}{x \cdot x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.5%

      \[\leadsto \frac{\frac{-1}{x}}{\color{blue}{x}} \]

   if 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x - 1\right)\right)} - \frac{1}{x} \]
   4. Step-by-step derivation

      1. distribute-rgt-out-- N/A

         \[\leadsto \left(1 + \color{blue}{\left(x \cdot x - 1 \cdot x\right)}\right) - \frac{1}{x} \]

      2. unpow2 N/A

         \[\leadsto \left(1 + \left(\color{blue}{{x}^{2}} - 1 \cdot x\right)\right) - \frac{1}{x} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \left(1 + \color{blue}{\left({x}^{2} + \left(\mathsf{neg}\left(1\right)\right) \cdot x\right)}\right) - \frac{1}{x} \]

      4. metadata-eval N/A

         \[\leadsto \left(1 + \left({x}^{2} + \color{blue}{-1} \cdot x\right)\right) - \frac{1}{x} \]

      5. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(1 + {x}^{2}\right) + -1 \cdot x\right)} - \frac{1}{x} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\left(1 + {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) - \frac{1}{x} \]

      7. unsub-neg N/A

         \[\leadsto \color{blue}{\left(\left(1 + {x}^{2}\right) - x\right)} - \frac{1}{x} \]

      8. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + {x}^{2}\right) - x\right)} - \frac{1}{x} \]

      9. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left({x}^{2} + 1\right)} - x\right) - \frac{1}{x} \]

      10. unpow2 N/A

          \[\leadsto \left(\left(\color{blue}{x \cdot x} + 1\right) - x\right) - \frac{1}{x} \]

      11. lower-fma.f64 99.1

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(x, x, 1\right)} - x\right) - \frac{1}{x} \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, x, 1\right) - x\right)} - \frac{1}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} + \frac{-1}{x} \leq -500:\\ \;\;\;\;\left(x + -1\right) \cdot \left(x + \frac{1}{x}\right)\\ \mathbf{elif}\;\frac{1}{1 + x} + \frac{-1}{x} \leq 0:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, x, 1\right) - x\right) + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x} + \frac{-1}{x}\\ t_1 := \left(x + -1\right) \cdot \left(x + \frac{1}{x}\right)\\ \mathbf{if}\;t\_0 \leq -500:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x)))
        (t_1 (* (+ x -1.0) (+ x (/ 1.0 x)))))
   (if (<= t_0 -500.0) t_1 (if (<= t_0 0.0) (/ (/ -1.0 x) x) t_1))))

C

double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double t_1 = (x + -1.0) * (x + (1.0 / x));
	double tmp;
	if (t_0 <= -500.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 / x) / x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 / (1.0d0 + x)) + ((-1.0d0) / x)
    t_1 = (x + (-1.0d0)) * (x + (1.0d0 / x))
    if (t_0 <= (-500.0d0)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = ((-1.0d0) / x) / x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double t_1 = (x + -1.0) * (x + (1.0 / x));
	double tmp;
	if (t_0 <= -500.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 / x) / x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x)
	t_1 = (x + -1.0) * (x + (1.0 / x))
	tmp = 0
	if t_0 <= -500.0:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = (-1.0 / x) / x
	else:
		tmp = t_1
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-1.0 / x))
	t_1 = Float64(Float64(x + -1.0) * Float64(x + Float64(1.0 / x)))
	tmp = 0.0
	if (t_0 <= -500.0)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-1.0 / x) / x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	t_1 = (x + -1.0) * (x + (1.0 / x));
	tmp = 0.0;
	if (t_0 <= -500.0)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = (-1.0 / x) / x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + -1.0), $MachinePrecision] * N[(x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -500.0], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < -500 or 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 + x \cdot \left(x - 1\right)\right) - 1}{x}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot \left(1 + x \cdot \left(x - 1\right)\right)}{x} - \frac{1}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(x - 1\right)\right) \cdot x}}{x} - \frac{1}{x} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(1 + x \cdot \left(x - 1\right)\right) \cdot \frac{x}{x}} - \frac{1}{x} \]

      4. *-inverses N/A

         \[\leadsto \left(1 + x \cdot \left(x - 1\right)\right) \cdot \color{blue}{1} - \frac{1}{x} \]

      5. *-rgt-identity N/A

         \[\leadsto \color{blue}{\left(1 + x \cdot \left(x - 1\right)\right)} - \frac{1}{x} \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x - 1\right) + 1\right)} - \frac{1}{x} \]

      7. associate--l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(x - 1\right) + \left(1 - \frac{1}{x}\right)} \]

      8. sub-neg N/A

         \[\leadsto x \cdot \left(x - 1\right) + \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto x \cdot \left(x - 1\right) + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) + 1\right)} \]

      10. neg-sub0 N/A

          \[\leadsto x \cdot \left(x - 1\right) + \left(\color{blue}{\left(0 - \frac{1}{x}\right)} + 1\right) \]

      11. associate-+l- N/A

          \[\leadsto x \cdot \left(x - 1\right) + \color{blue}{\left(0 - \left(\frac{1}{x} - 1\right)\right)} \]

      12. neg-sub0 N/A

          \[\leadsto x \cdot \left(x - 1\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{x} - 1\right)\right)\right)} \]

      13. unsub-neg N/A

          \[\leadsto \color{blue}{x \cdot \left(x - 1\right) - \left(\frac{1}{x} - 1\right)} \]

      14. *-inverses N/A

          \[\leadsto x \cdot \left(x - 1\right) - \left(\frac{1}{x} - \color{blue}{\frac{x}{x}}\right) \]

      15. div-sub N/A

          \[\leadsto x \cdot \left(x - 1\right) - \color{blue}{\frac{1 - x}{x}} \]

      16. unsub-neg N/A

          \[\leadsto x \cdot \left(x - 1\right) - \frac{\color{blue}{1 + \left(\mathsf{neg}\left(x\right)\right)}}{x} \]

      17. mul-1-neg N/A

          \[\leadsto x \cdot \left(x - 1\right) - \frac{1 + \color{blue}{-1 \cdot x}}{x} \]

      18. *-rgt-identity N/A

          \[\leadsto x \cdot \left(x - 1\right) - \frac{\color{blue}{\left(1 + -1 \cdot x\right) \cdot 1}}{x} \]

      19. associate-/l* N/A

          \[\leadsto x \cdot \left(x - 1\right) - \color{blue}{\left(1 + -1 \cdot x\right) \cdot \frac{1}{x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\left(-1 + x\right) \cdot \left(x + \frac{1}{x}\right)} \]

   if -500 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 59.2%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-1}{\color{blue}{x \cdot x}} \]

      3. lower-*.f64 98.2

         \[\leadsto \frac{-1}{\color{blue}{x \cdot x}} \]

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\frac{-1}{x \cdot x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.5%

      \[\leadsto \frac{\frac{-1}{x}}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} + \frac{-1}{x} \leq -500:\\ \;\;\;\;\left(x + -1\right) \cdot \left(x + \frac{1}{x}\right)\\ \mathbf{elif}\;\frac{1}{1 + x} + \frac{-1}{x} \leq 0:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -1\right) \cdot \left(x + \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x} + \frac{-1}{x}\\ t_1 := \left(x + -1\right) \cdot \left(x + \frac{1}{x}\right)\\ \mathbf{if}\;t\_0 \leq -500:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x)))
        (t_1 (* (+ x -1.0) (+ x (/ 1.0 x)))))
   (if (<= t_0 -500.0) t_1 (if (<= t_0 0.0) (/ -1.0 (* x x)) t_1))))

C

double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double t_1 = (x + -1.0) * (x + (1.0 / x));
	double tmp;
	if (t_0 <= -500.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 / (1.0d0 + x)) + ((-1.0d0) / x)
    t_1 = (x + (-1.0d0)) * (x + (1.0d0 / x))
    if (t_0 <= (-500.0d0)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double t_1 = (x + -1.0) * (x + (1.0 / x));
	double tmp;
	if (t_0 <= -500.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x)
	t_1 = (x + -1.0) * (x + (1.0 / x))
	tmp = 0
	if t_0 <= -500.0:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = -1.0 / (x * x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-1.0 / x))
	t_1 = Float64(Float64(x + -1.0) * Float64(x + Float64(1.0 / x)))
	tmp = 0.0
	if (t_0 <= -500.0)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	t_1 = (x + -1.0) * (x + (1.0 / x));
	tmp = 0.0;
	if (t_0 <= -500.0)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = -1.0 / (x * x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + -1.0), $MachinePrecision] * N[(x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -500.0], t$95$1, If[LessEqual[t$95$0, 0.0], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < -500 or 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 + x \cdot \left(x - 1\right)\right) - 1}{x}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot \left(1 + x \cdot \left(x - 1\right)\right)}{x} - \frac{1}{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(x - 1\right)\right) \cdot x}}{x} - \frac{1}{x} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(1 + x \cdot \left(x - 1\right)\right) \cdot \frac{x}{x}} - \frac{1}{x} \]

      4. *-inverses N/A

         \[\leadsto \left(1 + x \cdot \left(x - 1\right)\right) \cdot \color{blue}{1} - \frac{1}{x} \]

      5. *-rgt-identity N/A

         \[\leadsto \color{blue}{\left(1 + x \cdot \left(x - 1\right)\right)} - \frac{1}{x} \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x - 1\right) + 1\right)} - \frac{1}{x} \]

      7. associate--l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(x - 1\right) + \left(1 - \frac{1}{x}\right)} \]

      8. sub-neg N/A

         \[\leadsto x \cdot \left(x - 1\right) + \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto x \cdot \left(x - 1\right) + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) + 1\right)} \]

      10. neg-sub0 N/A

          \[\leadsto x \cdot \left(x - 1\right) + \left(\color{blue}{\left(0 - \frac{1}{x}\right)} + 1\right) \]

      11. associate-+l- N/A

          \[\leadsto x \cdot \left(x - 1\right) + \color{blue}{\left(0 - \left(\frac{1}{x} - 1\right)\right)} \]

      12. neg-sub0 N/A

          \[\leadsto x \cdot \left(x - 1\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{x} - 1\right)\right)\right)} \]

      13. unsub-neg N/A

          \[\leadsto \color{blue}{x \cdot \left(x - 1\right) - \left(\frac{1}{x} - 1\right)} \]

      14. *-inverses N/A

          \[\leadsto x \cdot \left(x - 1\right) - \left(\frac{1}{x} - \color{blue}{\frac{x}{x}}\right) \]

      15. div-sub N/A

          \[\leadsto x \cdot \left(x - 1\right) - \color{blue}{\frac{1 - x}{x}} \]

      16. unsub-neg N/A

          \[\leadsto x \cdot \left(x - 1\right) - \frac{\color{blue}{1 + \left(\mathsf{neg}\left(x\right)\right)}}{x} \]

      17. mul-1-neg N/A

          \[\leadsto x \cdot \left(x - 1\right) - \frac{1 + \color{blue}{-1 \cdot x}}{x} \]

      18. *-rgt-identity N/A

          \[\leadsto x \cdot \left(x - 1\right) - \frac{\color{blue}{\left(1 + -1 \cdot x\right) \cdot 1}}{x} \]

      19. associate-/l* N/A

          \[\leadsto x \cdot \left(x - 1\right) - \color{blue}{\left(1 + -1 \cdot x\right) \cdot \frac{1}{x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\left(-1 + x\right) \cdot \left(x + \frac{1}{x}\right)} \]

   if -500 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 59.2%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-1}{\color{blue}{x \cdot x}} \]

      3. lower-*.f64 98.2

         \[\leadsto \frac{-1}{\color{blue}{x \cdot x}} \]

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\frac{-1}{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} + \frac{-1}{x} \leq -500:\\ \;\;\;\;\left(x + -1\right) \cdot \left(x + \frac{1}{x}\right)\\ \mathbf{elif}\;\frac{1}{1 + x} + \frac{-1}{x} \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -1\right) \cdot \left(x + \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x} + \frac{-1}{x}\\ t_1 := \left(1 - x\right) + \frac{-1}{x}\\ \mathbf{if}\;t\_0 \leq -500:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x))) (t_1 (+ (- 1.0 x) (/ -1.0 x))))
   (if (<= t_0 -500.0) t_1 (if (<= t_0 0.0) (/ -1.0 (* x x)) t_1))))

C

double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double t_1 = (1.0 - x) + (-1.0 / x);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 / (1.0d0 + x)) + ((-1.0d0) / x)
    t_1 = (1.0d0 - x) + ((-1.0d0) / x)
    if (t_0 <= (-500.0d0)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double t_1 = (1.0 - x) + (-1.0 / x);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x)
	t_1 = (1.0 - x) + (-1.0 / x)
	tmp = 0
	if t_0 <= -500.0:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = -1.0 / (x * x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-1.0 / x))
	t_1 = Float64(Float64(1.0 - x) + Float64(-1.0 / x))
	tmp = 0.0
	if (t_0 <= -500.0)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	t_1 = (1.0 - x) + (-1.0 / x);
	tmp = 0.0;
	if (t_0 <= -500.0)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = -1.0 / (x * x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -500.0], t$95$1, If[LessEqual[t$95$0, 0.0], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < -500 or 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + -1 \cdot x\right)} - \frac{1}{x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) - \frac{1}{x} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\left(1 - x\right)} - \frac{1}{x} \]

      3. lower--.f64 99.0

         \[\leadsto \color{blue}{\left(1 - x\right)} - \frac{1}{x} \]

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\left(1 - x\right)} - \frac{1}{x} \]

   if -500 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 59.2%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-1}{\color{blue}{x \cdot x}} \]

      3. lower-*.f64 98.2

         \[\leadsto \frac{-1}{\color{blue}{x \cdot x}} \]

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\frac{-1}{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} + \frac{-1}{x} \leq -500:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \mathbf{elif}\;\frac{1}{1 + x} + \frac{-1}{x} \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + x} + \frac{-1}{x}\\ t_1 := 1 + \frac{-1}{x}\\ \mathbf{if}\;t\_0 \leq -500:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x))) (t_1 (+ 1.0 (/ -1.0 x))))
   (if (<= t_0 -500.0) t_1 (if (<= t_0 0.0) (/ -1.0 (* x x)) t_1))))

C

double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double t_1 = 1.0 + (-1.0 / x);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 / (1.0d0 + x)) + ((-1.0d0) / x)
    t_1 = 1.0d0 + ((-1.0d0) / x)
    if (t_0 <= (-500.0d0)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	double t_1 = 1.0 + (-1.0 / x);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = -1.0 / (x * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x)
	t_1 = 1.0 + (-1.0 / x)
	tmp = 0
	if t_0 <= -500.0:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = -1.0 / (x * x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-1.0 / x))
	t_1 = Float64(1.0 + Float64(-1.0 / x))
	tmp = 0.0
	if (t_0 <= -500.0)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(-1.0 / Float64(x * x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (1.0 / (1.0 + x)) + (-1.0 / x);
	t_1 = 1.0 + (-1.0 / x);
	tmp = 0.0;
	if (t_0 <= -500.0)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = -1.0 / (x * x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -500.0], t$95$1, If[LessEqual[t$95$0, 0.0], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < -500 or 0.0 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x))

   1. Initial program 100.0%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} - \frac{1}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.3%

      \[\leadsto \color{blue}{1} - \frac{1}{x} \]

   if -500 < (-.f64 (/.f64 #s(literal 1 binary64) (+.f64 x #s(literal 1 binary64))) (/.f64 #s(literal 1 binary64) x)) < 0.0

   1. Initial program 59.2%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-1}{\color{blue}{x \cdot x}} \]

      3. lower-*.f64 98.2

         \[\leadsto \frac{-1}{\color{blue}{x \cdot x}} \]

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\frac{-1}{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + x} + \frac{-1}{x} \leq -500:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;\frac{1}{1 + x} + \frac{-1}{x} \leq 0:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{-1}{x}}{x}\\ \mathbf{if}\;x \leq -380000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 200000000:\\ \;\;\;\;\frac{1}{1 + x} + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (/ -1.0 x) x)))
   (if (<= x -380000000.0)
     t_0
     (if (<= x 200000000.0) (+ (/ 1.0 (+ 1.0 x)) (/ -1.0 x)) t_0))))

C

double code(double x) {
	double t_0 = (-1.0 / x) / x;
	double tmp;
	if (x <= -380000000.0) {
		tmp = t_0;
	} else if (x <= 200000000.0) {
		tmp = (1.0 / (1.0 + x)) + (-1.0 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / x) / x
    if (x <= (-380000000.0d0)) then
        tmp = t_0
    else if (x <= 200000000.0d0) then
        tmp = (1.0d0 / (1.0d0 + x)) + ((-1.0d0) / x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (-1.0 / x) / x;
	double tmp;
	if (x <= -380000000.0) {
		tmp = t_0;
	} else if (x <= 200000000.0) {
		tmp = (1.0 / (1.0 + x)) + (-1.0 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (-1.0 / x) / x
	tmp = 0
	if x <= -380000000.0:
		tmp = t_0
	elif x <= 200000000.0:
		tmp = (1.0 / (1.0 + x)) + (-1.0 / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(-1.0 / x) / x)
	tmp = 0.0
	if (x <= -380000000.0)
		tmp = t_0;
	elseif (x <= 200000000.0)
		tmp = Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-1.0 / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (-1.0 / x) / x;
	tmp = 0.0;
	if (x <= -380000000.0)
		tmp = t_0;
	elseif (x <= 200000000.0)
		tmp = (1.0 / (1.0 + x)) + (-1.0 / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -380000000.0], t$95$0, If[LessEqual[x, 200000000.0], N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8e8 or 2e8 < x

   1. Initial program 58.8%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-1}{\color{blue}{x \cdot x}} \]

      3. lower-*.f64 99.3

         \[\leadsto \frac{-1}{\color{blue}{x \cdot x}} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{-1}{x \cdot x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.6%

      \[\leadsto \frac{\frac{-1}{x}}{\color{blue}{x}} \]

   if -3.8e8 < x < 2e8

   1. Initial program 99.4%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -380000000:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{elif}\;x \leq 200000000:\\ \;\;\;\;\frac{1}{1 + x} + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -1.0 x))

C

double code(double x) {
	return -1.0 / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) / x
end function

Java

public static double code(double x) {
	return -1.0 / x;
}

Python

def code(x):
	return -1.0 / x

Julia

function code(x)
	return Float64(-1.0 / x)
end

MATLAB

function tmp = code(x)
	tmp = -1.0 / x;
end

Wolfram

code[x_] := N[(-1.0 / x), $MachinePrecision]

Derivation

1. Initial program 78.8%

   \[\frac{1}{x + 1} - \frac{1}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-1}{x}} \]
4. Step-by-step derivation

   1. lower-/.f64 49.6

      \[\leadsto \color{blue}{\frac{-1}{x}} \]

5. Applied rewrites 49.6%

   \[\leadsto \color{blue}{\frac{-1}{x}} \]
6. Add Preprocessing

Alternative 11: 2.9% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 78.8%

   \[\frac{1}{x + 1} - \frac{1}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{x \cdot \left(1 + x \cdot \left(x - 1\right)\right) - 1}{x}} \]
4. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 + x \cdot \left(x - 1\right)\right)}{x} - \frac{1}{x}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(x - 1\right)\right) \cdot x}}{x} - \frac{1}{x} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x - 1\right)\right) \cdot \frac{x}{x}} - \frac{1}{x} \]

   4. *-inverses N/A

      \[\leadsto \left(1 + x \cdot \left(x - 1\right)\right) \cdot \color{blue}{1} - \frac{1}{x} \]

   5. *-rgt-identity N/A

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x - 1\right)\right)} - \frac{1}{x} \]

   6. +-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(x - 1\right) + 1\right)} - \frac{1}{x} \]

   7. associate--l+ N/A

      \[\leadsto \color{blue}{x \cdot \left(x - 1\right) + \left(1 - \frac{1}{x}\right)} \]

   8. sub-neg N/A

      \[\leadsto x \cdot \left(x - 1\right) + \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right)} \]

   9. +-commutative N/A

      \[\leadsto x \cdot \left(x - 1\right) + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right) + 1\right)} \]

   10. neg-sub0 N/A

       \[\leadsto x \cdot \left(x - 1\right) + \left(\color{blue}{\left(0 - \frac{1}{x}\right)} + 1\right) \]

   11. associate-+l- N/A

       \[\leadsto x \cdot \left(x - 1\right) + \color{blue}{\left(0 - \left(\frac{1}{x} - 1\right)\right)} \]

   12. neg-sub0 N/A

       \[\leadsto x \cdot \left(x - 1\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{x} - 1\right)\right)\right)} \]

   13. unsub-neg N/A

       \[\leadsto \color{blue}{x \cdot \left(x - 1\right) - \left(\frac{1}{x} - 1\right)} \]

   14. *-inverses N/A

       \[\leadsto x \cdot \left(x - 1\right) - \left(\frac{1}{x} - \color{blue}{\frac{x}{x}}\right) \]

   15. div-sub N/A

       \[\leadsto x \cdot \left(x - 1\right) - \color{blue}{\frac{1 - x}{x}} \]

   16. unsub-neg N/A

       \[\leadsto x \cdot \left(x - 1\right) - \frac{\color{blue}{1 + \left(\mathsf{neg}\left(x\right)\right)}}{x} \]

   17. mul-1-neg N/A

       \[\leadsto x \cdot \left(x - 1\right) - \frac{1 + \color{blue}{-1 \cdot x}}{x} \]

   18. *-rgt-identity N/A

       \[\leadsto x \cdot \left(x - 1\right) - \frac{\color{blue}{\left(1 + -1 \cdot x\right) \cdot 1}}{x} \]

   19. associate-/l* N/A

       \[\leadsto x \cdot \left(x - 1\right) - \color{blue}{\left(1 + -1 \cdot x\right) \cdot \frac{1}{x}} \]

5. Applied rewrites 48.5%

   \[\leadsto \color{blue}{\left(-1 + x\right) \cdot \left(x + \frac{1}{x}\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto {x}^{2} \cdot \color{blue}{\left(\left(1 + \frac{1}{{x}^{2}}\right) - \frac{1}{x}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 2.4%

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{x}, 1 - x\right) \]

9. Taylor expanded in x around 0

   \[\leadsto 1 \]
10. Step-by-step derivation

11. Applied rewrites 3.1%

    \[\leadsto 1 \]
12. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-1}{x}}{x + 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ -1.0 x) (+ x 1.0)))

C

double code(double x) {
	return (-1.0 / x) / (x + 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-1.0d0) / x) / (x + 1.0d0)
end function

Java

public static double code(double x) {
	return (-1.0 / x) / (x + 1.0);
}

Python

def code(x):
	return (-1.0 / x) / (x + 1.0)

Julia

function code(x)
	return Float64(Float64(-1.0 / x) / Float64(x + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = (-1.0 / x) / (x + 1.0);
end

Wolfram

code[x_] := N[(N[(-1.0 / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

Developer Target 2: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot \left(-1 - x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (* x (- -1.0 x))))

C

double code(double x) {
	return 1.0 / (x * (-1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (x * ((-1.0d0) - x))
end function

Java

public static double code(double x) {
	return 1.0 / (x * (-1.0 - x));
}

Python

def code(x):
	return 1.0 / (x * (-1.0 - x))

Julia

function code(x)
	return Float64(1.0 / Float64(x * Float64(-1.0 - x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (x * (-1.0 - x));
end

Wolfram

code[x_] := N[(1.0 / N[(x * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024238 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ -1 x) (+ x 1)))

  :alt
  (! :herbie-platform default (/ 1 (* x (- -1 x))))

  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))